I experimentally discovered the following conjectures: $$\Re\Big[1800\operatorname{Li}_3(i\,\phi)-24\operatorname{Li}_3\left(i\,\phi^5\right)\Big]\stackrel{\color{gray}?}=100\ln^3\phi-47\,\pi^2\ln\phi-150\,\zeta(3),\tag1$$ $$\Im\Big[720\operatorname{Li}_3(i\,\phi)-320\operatorname{Li}_3\left(i\,\phi^3\right)-48\operatorname{Li}_3\left(i\,\phi^5\right)\Big]\stackrel{\color{gray}?}=9\,\pi^3-780\,\pi\ln^2\phi,\tag2$$ where $\phi=\frac{1+\sqrt5}2$ is the golden ratio and $\operatorname{Li}_3(z)$ is the trilogarithm. They check numerically with at least $20000$ decimal digits. It appears that Maple and Mathematica know nothing about these identities.
Are these known identities? How can we prove them?
Update: It also appears that $$\begin{align}&\Re\operatorname{Li}_3(i\,\phi)\stackrel{\color{gray}?}=\frac1{32}\operatorname{Li}_3\left(\phi^{-4}\right)+\frac3{16}\operatorname{Li}_3\left(\phi^{-2}\right)-\frac38\ln^3\phi-\frac14\zeta(3)\\ \,\\ &\Re\operatorname{Li}_3\left(i\,\phi^3\right)\stackrel{\color{gray}?}=\frac9{32}\operatorname{Li}_3\left(\phi^{-4}\right)+\frac12\operatorname{Li}_3\left(\phi^{-3}\right)+\frac38\operatorname{Li}_3\left(\phi^{-2}\right)-\frac38\operatorname{Li}_3\left(\phi^{-1}\right)-\frac{43}8\ln^3\phi-\frac{15}{32}\zeta(3)\\ \,\\ &\Re\operatorname{Li}_3\left(i\,\phi^5\right)\stackrel{\color{gray}?}=\frac{75}{32}\operatorname{Li}_3\left(\phi^{-4}\right)-\frac58\operatorname{Li}_3\left(\phi^{-2}\right)-\frac{45}2\ln^3\phi-\frac34\zeta(3)\end{align}$$ These together with a known value for $\operatorname{Li}_3\left(\phi^{-2}\right)$ imply $(1)$.